We prove Theorem REF  by combining the multilinear Kakeya theorem [1]}, [2]} with the multilinear to linear arguments developed by Bourgain and Guth in [3]}; this is done in Section . A secondary goal of this paper is provide a short and self-contained introduction to this type of argument. We prove Conjecture REF  by first using a result from [4]} to reduce to the case where \(L\)  is analytic and hence supports a Frostman measure, and then using a standard discretization argument; this is done in Section .
